In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (− Δ ) s in ℝ n , for n ≥ 2, such as (0.1) (−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1. $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q∈2,2s⋆ $q \in\left[2,2_{s}^{\star}\right)$with 2s⋆:=2nn−2s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$being the fractional critical Sobolev exponent, E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f ( u ) is a continuous function on ℝ of quasi-critical, super- q -linear growth with f ( u ) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) (−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1, $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces M s ; q , p (ℝ n ) as well as their associated compact embedding results.

中文翻译：

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紧凑的 Sobolev-Slobodeckij 嵌入和分数拉普拉斯方程的正解

在这项工作中，我们研究了一个椭圆方程的正解的存在，该方程涉及 ℝ n 中的分数拉普拉斯算子 (− Δ ) s，对于 n ≥ 2，例如 (0.1) (−Δ)su+E(x)u +V(x)uq−1=K(x)f(u)+u2s⋆−1。$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\ star}-1}.$$ 这里，s ∈ (0, 1), q∈2,2s⋆$q \in\left[2,2_{s}^{\star}\right)$with 2s⋆： =2nn−2s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$为分数临界Sobolev指数，E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ 是可测函数，在测度理论意义上满足联合“在无穷远处消失”条件，并且 f ( u ) 是 ℝ 上的一个连续函数，具有 f 的准临界、超 q 线性增长( u ) ≥ 0 如果 u ≥ 0。此外，我们研究了 ℝ n 中椭圆方程的多个正解的存在性，例如 (0.2) (−Δ)su+E(x)u+V(x)uq− 1=λK(x)ur−1, $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ 其中 2 < r < q < ∞（都可能（超）临界），E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ 是满足联合可积性条件的可测函数，λ > 0 是范围。为了研究 (0.1)-(0.2)，我们首先描述了一个一般分数 Sobolev-Slobodeckij 空间 M s 的族；q , p (ℝ n ) 以及它们相关的紧凑嵌入结果。